📘 complex algebra

1. The problem is to solve the equation $$\left(z - \frac{1}{\sqrt{2}}\right)^{105} + \left(z + \frac{1}{\sqrt{2}}\right)^{105} = 0$$ where $z \in \mathbb{C}$. 2. This is a complex

1. **State the problem:** Solve the equation $z^* = z^3$ where $z^*$ denotes the complex conjugate of $z$. 2. **Recall definitions and formulas:** For a complex number $z = x + yi$

1. Problem 3.1: Find all complex numbers $z$ satisfying $ (1 + 2i)(i - z) + (3 - 4i)(1 - iz) = 1 + 7i$. 2. Use distributive property and simplify each product:

1. **State the problem:** Given the complex number equation $\frac{\overline{Z}}{Z} = \frac{3}{5} + \frac{1}{5}i$ and the condition $Z + \overline{Z} = 4$, find the complex number

1. Statement of the problem: Solve $ (i - x)^3 = -1$. 2. Formula and rule: For complex numbers, to solve $z^3 = w$ we write $w$ in polar form and take cube roots using $$z = e^{i(\

1. **State the problem:** We want to simplify the complex expression

1. **Graph the complex numbers on the complex plane**. Each complex number $a + bi$ is represented as the point $(a,b)$ where $a$ is the real part (x-axis) and $b$ is the imaginary

1. Solve the system: (i) $z - 4w = 3i$ and $2z + 3w = 11 - 5i$. Multiply first eq by 3: $3z - 12w = 9i$

1. **Solve the system (i):** Given:

1. We are given the equation $$|z| - z = 1 + 2i$$ and need to find the value of complex number $$z$$. 2. Let $$z = x + yi$$ where $$x, y \in \mathbb{R}$$.

1. **State the problem:** Find all complex numbers $z$ such that $$ (z+1)^3 = (\overline{z} - 2 + \sqrt{3}i)^3. $$ 2. **Apply the cube root:** Since both sides are cubes, we equate